Stochastic Differential Equations: A SAD Primer

1 The Wiener Process The stochastic process W t is a Wiener Process if its increments are normally distributed: W t − W s ∼ N (0, t − s) (1) and are independent if non-overlapping: E{(W t 1 − W s 1)(W t 2 − W s 2)} = 0 if (s 1 , t 1) ∩ (s 2 , t 2) = ∅ (2) In particular, we have E{W t − W s } = 0 (3) E{(W t − W s) 2 } = t − s (4) Furthermore, if we fix the initial value of the process to be zero, W 0 = 0 (5) then we obtain E{W t } = 0 (6) E{(W t) 2 } = t (7) E{W t W s } = min(t, s) (8) where the third of these equations follows from (2) by noting that if t > s , then W t = (W t − W s) + W s (9) so E{W t W s } = E{((W t − W s) + W s)W s } (10) = E{(W t − W s)W s } + E{(W s) 2 } (11) = 0 + s (12) by linearity of the expectation operator. Realisations of the Wiener process are a.s. continuous, ie, P(lim s→t X s = X t) = 1 (13) but are nowhere differentiable. This latter result can be understood to arise from the fact that W t +h − W t h ∼ N 0, 1 h (14) which distribution has divergent variance in the limit h → 0.